What can I do to help my brother and dad's deteriorating relationship

Butt out.

I know you're going to resist this advice.

Follow it anyway.

Everyone is an adult, yes?

Then it's up to your brother if he wants to fight any battles with your folks, or whatever are his conditions for forgiveness or whatever.

If your folks bad mouth your brother, your response is, "He's my brother and I love him. Please stop talking about him this way."

Then you change the subject. If that doesn't work, then you leave the room.

Lather, rinse, repeat.

If your folks demand some sort of an explanation, then tell them it's unfair for them to put you in the middle of things. "And here's his number. Please call him if you feel the need to talk about him and leave me out of it."

Then get your jacket and go for a walk or a drive. Remove yourself from the situation.

Will they make up?

Who the hell knows?

But in the meantime, understand that everyone is behaving childishly. And you can love all of the parties in this equation without having to play mediator. You also don't have to be a repository for hostility -- so if your brother bad mouths your folks, you can say the same to him.

And no matter what, it is not on you to fix any of this. Will they regret their behavior? Probably. Again, not your call to rush in like a white knight and fix them.